Let $f$ be a transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$. Its Jacobian matrix is given below. $J(f) = \begin{bmatrix} 8x & 8y \\ \\ 3y & 3x \end{bmatrix}$ Find the Jacobian determinant of $f$. $|J(f)| = $ How will $f$ expand or contract space around the point $(2, 3)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A Leave it the same (Choice B) B Expand it finitely (Choice C) C Contract it finitely (Choice D) D Contract it infinitely
Explanation: The Jacobian determinant is the determinant of the Jacobian matrix. It represents the factor by which the transformation $f$ expands or contracts volume around a certain input. $\begin{aligned} |J(f)| &= \det \left( \begin{bmatrix} 8x & 8y \\ \\ 3y & 3x \end{bmatrix} \right) \\ \\ &=24x^2 - 24y^2 \\ \\ &= 24(x^2 - y^2) \end{aligned}$ If we evaluate $|J(f)|$ at $(2, 3)$, we get $-120$. Because the Jacobian determinant here has an absolute value greater than $1$, we can conclude that $f$ will finitely expand the space around $(2, 3)$. To recap, the Jacobian determinant of $f$ is $24(x^2 - y^2)$, and $f$ will finitely expand the space around the point $(2, 3)$.